PrimitiveIndResults071302 - 1 PRIMITIVE INDEPENDENCE...

1 PRIMITIVE INDEPENDENCE RESULTS by Harvey M. Friedman Department of Mathematics Ohio State University July 13, 2002 Abstract. We present some new set and class theoretic independence results from ZFC and NBGC that are particularly simple and close to the primitives of membership and equality (see sections 4,5). They are shown to be equivalent to familiar small large cardinal hypotheses. We modify these independendent statements in order to give an example of a sentence in set theory with 5 quantifiers which is independent of ZFC (see section 6). It is known that all 3 quantifier sentences are decided in a weak fragment of ZF without power set (see [Fr02a]). 1. SUBTLE CARDINALS. Subtle cardinals were first defined in a 1971 unpublished paper of Ronald Jensen and Ken Kunen. The subtle cardinal hierarchy was first presented in [Ba75]. The main results of [Ba75] were reworked in [Fr01]. [Fr01] also presents a number of new properties of ordinals (and linear orderings) not mentioning closed unbounded sets, which correspond to the subtle cardinal hierarchy. The new properties from [Fr01] are not quite in the right form to be applied directly to this context. We need to use some new properties - particularly a property called weakly inclusion subtle. We follow the usual set theoretic convention of taking ordinals to be epsilon connected transitive sets. The following definition is used in [Ba75] and [Fr01]. We say that an ordinal l is subtle if and only if i) l is a limit ordinal; ii) Let C l be closed unbounded, and for each a < l let A a a be given. There exists a , b C, a < b , such that A a = A b « a .

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2 We need the following new definition for present purposes. We say that an ordinal l is inclusion subtle if and only if i) l is a limit ordinal; ii) Let C l be closed unbounded, and for each a < l let A a a be given. There exists a , b C, a < b , such that A a A b . LEMMA 1.1. Every inclusion subtle ordinal is an uncountable cardinal. Proof: By setting A0 = ∅ and A n+1 = {n}, we see that w is not inclusion subtle. Let l be inclusion subtle and not a cardinal. Let h: l Æ d be one-one, where d < l is the cardinal of l . Then d ≥ w . Define A a = {h( a )} for d < a < l , A a = ∅ otherwise. Let C = ( d , l ). This is a counterexample to the inclusion subtlety of l . QED In light of Lemma 1.1, we drop the terminology “subtle ordinal” in favor of “subtle cardinal”. THEOREM 1.2. A cardinal is subtle if and only if it is inclusion subtle. Proof: Let l be inclusion subtle. Let C l be closed unbounded, and A a a , a < l , be given. Since l is an uncountable cardinal, we can assume that every element of C is a limit ordinal. For a C, define B a = {2 g : g A a } {2 g +1 < a : g A a }. Let a , b C, a < b , B a B b . Then A a = A b « a . Here 2 g is g copies of 2. QED For our purposes, we are particularly interested in the following somewhat technical notion.

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